Question

What is the remainder when 13^400 is divided by 11?

Answer 1

13/11 leaves a remainder 2 
Hence (13^400/11 == Remainder(2^400/11)

By remainder theorem(fermat's) :
remainder [M^(fi (n)) / N] = 1, where M, N are co-primes
and (fi(n)) is given by: N*(1 - 1/N)
Here, M=2 , N=11
(fi(n)) = 11* ( 1 - 1/11) = 11*10/11 = 10
So, Rem[ 2^10/11] = 1
2^400 = (2^10)^40
We kow 2^10/11 leaves a remainder 1, hence 2^400 too leaves a remainder 1
So 13^400 when divided by 11 leaves a remainder '1'
Hope it helps.